From: voloch@fireant.ma.utexas.edu (Felipe Voloch)
Subject: Re: a^x+....+a^z=b^n
Date: 19 Feb 1999 19:55:53 GMT
Newsgroups: sci.math.research
Keywords: S-unit equation; finitude of solutions to Sum(a^{x_i})=power

langlois bruno (blang@club-internet.fr) wrote:
: It seems that the following result is true :
: 
: any equation of the form a^x+a^y+...+a^z=b^n has only finitely many
: positive integer solutions (x,y,...,z,n) provided a and b are
: independent integers.
: 
: Can you give me some precisions (proof, correction ...) about this
: result ?
: 
: Thanks.
: 
: B.L.
: 

This is a consequence of a more general theorem on the "S-unit equation",
proved by Evertse, van der Poorten and Schlikewei. See

86c:11045 11J13 (11B37 11J86 11R27) 
Evertse, Jan-Hendrik(NL-LEID) 
On sums of $S$-units and linear recurrences. 
Compositio Math. 53 (1984), no. 2, 225--244. 


Felipe